Second, there exists an invariance of \(\pi _1\)  under a repetitive action of the Golden mean substitution (the Fibonacci map) \(\rho :a \rightarrow ab\) , \(b \rightarrow a\)  or under the Silver mean substitution \(\rho :a \rightarrow aba\) , \(b \rightarrow a\) . The terms Golden and Silver refer to the Perron-Frobenius eigenvalue of the substitution matrix [1]}.
Such an observation links the Hopf link, the group \(\pi _1\)  of the 2-torus and aperiodic substitutions.
